METHODS OF APPROXIMATION OF FUNCTIONS BY GENERALIZED POLYNOMIALS IN NUMERICAL ANALYSIS PROBLEMS RELATED TO CALCULATIONS ON APPROXIMATE DATA

Introduction: the methods of representation of functions given approximately by their singular integrals in relation to approximation problems and numerical methods for solving boundary value problems for differential equations are Investigated. In such a statement, the problem belongs to the class of ill-posed problems and requires the construction of appropriate regularizing methods and algorithms for its solution. Materials and methods of research: the questions of approximation of the function under study by generalized polynomials are Considered and some important properties of this computational process are investigated. The question of the extent to which the members of the approximating sequences represented by generalized polynomials "inherit" the analytical properties of the approximated functions is investigated. The nature and conditions of convergence of approximating sequences to the original function are investigated. Examples of application of the function approximation apparatus by generalized polynomials are given, calculation formulas of the computational algorithm are derived. Research results and their discussion: the results of theoretical studies obtained in the framework of this work are of independent importance in the field of computational mathematics. The practical significance of the results is that the proposed methods and approaches can be used in applied problems of the theory of approximation of functions, problems of applied analysis and boundary value problems of mathematical physics, using approximately given initial data obtained in the course of physical experiments or empirical functions.

аппроксимация функции, обобщенные полиномы, регуляризирующие алгоритмы, сингулярные интегралы, вычислительный процесс, задачи прикладного анализа, function approximation, generalized polynomials, regularizing algorithms, singular integrals, computational process, applied analysis problems

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